Calculating Displacement in a Coordinate System: A Practical Example

In this article, we'll explore the motion of a student who traverses east and south in a specific path. Using vector components and the principles of a coordinate system, we'll calculate the student's final displacement and discuss its direction.

Understanding the Problem

The student's journey consists of three distinct segments. First, the student moves 400 meters east. Next, the student moves 600 meters south. Lastly, the student moves 200 meters east. Our goal is to determine the student's final displacement from the starting point and the direction of this displacement.

Breaking Down the Movement

First Movement: The student moves 400 meters east. This can be represented in the coordinate system as (0, 0) -> (400, 0).

Second Movement: The student then moves 600 meters south. This changes the coordinates to (400, 0) -> (400, -600).

Third Movement: The student moves an additional 200 meters east. This brings the coordinates to (400, -600) -> (600, -600).

Total Movement and Resultant Vector

By summing up these movements, we can determine the student's total displacement:

Total Eastward Movement: 400 meters 200 meters 600 meters Total Southward Movement: 600 meters

These movements form a right triangle with the eastward and southward distances as the legs. To find the magnitude of the displacement, we can use the Pythagorean theorem:

Displacement  sqrt{(600 m)^2   (600 m)^2}

Calculating this, we get:

Displacement  sqrt{360000 m^2   360000 m^2}  sqrt{720000 m^2}  600 * sqrt{2} m ≈ 848.53 m

Direction of Displacement

The direction of the displacement can be determined by the angle it makes with the eastward direction. This angle, denoted as θ, can be found using the tangent function:

tanθ (opposite side) / (adjacent side) 600 m / 600 m 1

Therefore, θ 45 degrees, meaning the displacement is 45 degrees south of east.

Conclusion

The student's final displacement is approximately 848.53 meters in a direction 45 degrees south of east. This problem demonstrates the application of vector components and the coordinate system in resolving displacement.